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## Exponential Functions and Models

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**Exponential Functions and Models**Lesson 5.3**Definition**• An exponential function • Note the variable is in the exponent • The base is a • C is the coefficient, also considered the initial value (when x = 0)**Explore Exponentials**• Given f(1) = 3, for each unit increase in x, the output is multiplied by 1.5 • Determine the exponential function**Explore Exponentials**• Graph these exponentials • What do you think the coefficient C and the base a do to the appearance of the graphs?**Contrast Linear vs. Exponential**• Suppose you have a choice of two different jobs at graduation • Start at $30,000 with a 6% per year increase • Start at $40,000 with $1200 per year raise • Which should you choose? • One is linear growth • One is exponential growth**Which Job?**• How do we get each nextvalue for Option A? • When is Option A better? • When is Option B better? • Rate of increase a constant $1200 • Rate of increase changing • Percent of increase is a constant • Ratio of successive years is 1.06**Example**• Consider a savings account with compounded yearly income • You have $100 in the account • You receive 5% annual interest View completed table**Compounded Interest**• Completed table**Compounded Interest**• Table of results from calculator • Set Y= screen y1(x)=100*1.05^x • Choose Table (♦Y) • Graph of results**Assignment A**• Lesson 5.3A • Page 415 • Exercises 1 – 57 EOO**Compound Interest**• Consider an amount A0 of money deposited in an account • Pays annual rate of interest r percent • Compounded m times per year • Stays in the account n years • Then the resulting balance An**Exponential Modeling**• Population growth often modeled by exponential function • Half life of radioactive materials modeled by exponential function**Growth Factor**• Recall formulanew balance = old balance + 0.05 * old balance • Another way of writing the formulanew balance = 1.05 * old balance • Why equivalent? • Growth factor: 1 + interest rate as a fraction**Decreasing Exponentials**• Consider a medication • Patient takes 100 mg • Once it is taken, body filters medication out over period of time • Suppose it removes 15% of what is present in the blood stream every hour Fill in the rest of the table What is the growth factor?**Decreasing Exponentials**• Completed chart • Graph Growth Factor = 0.85 Note: when growth factor < 1, exponential is a decreasing function**Solving Exponential Equations Graphically**• For our medication example when does the amount of medication amount to less than 5 mg • Graph the functionfor 0 < t < 25 • Use the graph todetermine when**General Formula**• All exponential functions have the general format: • Where • A = initial value • B = growth rate • t = number of time periods**Typical Exponential Graphs**• When B > 1 • When B < 1**Using e As the Base**• We have used y = A * Bt • Consider letting B = ek • Then by substitution y = A * (ek)t • Recall B = (1 + r) (the growth factor) • It turns out that**Continuous Growth**• The constant k is called the continuous percent growth rate • For Q = a bt • k can be found by solving ek = b • Then Q = a ek*t • For positive a • if k > 0 then Q is an increasing function • if k < 0 then Q is a decreasing function**Continuous Growth**• For Q = a ek*t Assume a > 0 • k > 0 • k < 0**Continuous Growth**• For the functionwhat is thecontinuous growth rate? • The growth rate is the coefficient of t • Growth rate = 0.4 or 40% • Graph the function (predict what it looks like)**Converting Between Forms**• Change to the form Q = A*Bt • We know B = ek • Change to the form Q = A*ek*t • We know k = ln B (Why?)**Continuous Growth Rates**• May be a better mathematical model for some situations • Bacteria growth • Decrease of medicine in the bloodstream • Population growth of a large group**Example**• A population grows from its initial level of 22,000 people and grows at a continuous growth rate of 7.1% per year. • What is the formula P(t), the population in year t? • P(t) = 22000*e.071t • By what percent does the population increase each year (What is the yearly growth rate)? • Use b = ek**Assignment B**• Lesson 5.3B • Page 417 • Exercises 65 – 85 odd